Viscosity approximation method with Meir-Keeler contractions for common zero of accretive operators in Banach spaces

Jong Kyu Kim,Truong Minh Tuyen

doi:10.1186/s13663-014-0256-3

Abstract

The purpose of this paper is to introduce a new iteration by the combination of the viscosity approximation with Meir-Keeler contractions and proximal point algorithm for finding common zeros of a finite family of accretive operators in a Banach space with a uniformly Gâteaux differentiable norm. The results of this paper improve and extend corresponding well-known results by many others.

Highlights

Let E be a real Banach space and let J be the normalized duality mapping from E into E∗ given byJ(x) = f ∈ E∗ : x, f = x = f, ∀x ∈ E, where E∗ denotes the dual space of E and ·, · denotes the generalized duality pairing
Theorem . [ ] Let E be a strictly convex and reflexive Banach space with a uniformly Gâteaux differentiable norm, K be a nonempty, closed, and convex subset of E and Ai : K → E be an m-accretive operator, for each i =, . . . , N with
An accretive operator A defined on a Banach space E is said to satisfy the range condition if D(A) ⊂ R(I + λA), for all λ >, where D(A) denotes the closure of the domain of A

Summary

Introduction

[ ] Let E be a strictly convex and reflexive Banach space with a uniformly Gâteaux differentiable norm, K be a nonempty, closed, and convex subset of E and Ai : K → E be an m-accretive operator, for each i = , , . If every nonempty, bounded, closed, and convex subset of E has the fixed point property for nonexpansive mapping, {xn} converges strongly to a common solution of the equations

Objectives

Results

Conclusion